Contractibility of space of symplectically self-polar convex bodies

Abstract: Symplectically self-polar bodies have been introduced recently. They appeared in the context of Mahler's conjecture and in the context of symplectic outer billiard dynamics. In this paper we discuss the topology of the space of symplectically self-polar convex bodies. Namely, we show that this space is contractible.

• The paper proves that the space of fixed-dimensional symplectically self-polar convex bodies is contractible in the Hausdorff topology, establishing path-connectedness and eliminating topological component obstructions.
• The proof combines preservation under symplectic ℓ₂-sums with a reduction lemma and a continuously rotating family of isotropic subspaces to construct explicit homotopies between any two bodies.
• Planar examples show concrete homotopies between self-polar hexagons and the Euclidean disk, while leaving open whether this topology can help prove the higher-dimensional Mahler volume bound.

Overview

This paper by Mark Berezovik establishes a basic topological property of a recently introduced class of convex bodies: the space of symplectically self-polar convex bodies of fixed dimension, equipped with the Hausdorff topology, is contractible. The result is concise but closes a gap in the emerging theory of these bodies, which had previously been studied from the perspectives of Mahler-type volume inequalities and symplectic outer billiard dynamics, but not from the point of view of the topology of the class itself.

A convex body $X \subset \mathbb{R}^{2n}$ containing the origin in its interior is symplectically self-polar if $X = X^\omega$, where the symplectic polar is defined via the standard symplectic form $\omega$:

$$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$

The class was introduced in prior work of the author and collaborators, where it arose in two distinct contexts: first, as a reformulation of (symmetric) Mahler's conjecture — the statement that every symplectically self-polar body satisfies $\vol X \geq 2^n/n!$ is equivalent to Mahler's conjecture for centrally symmetric bodies; second, in dynamics, where such bodies furnish non-trivial examples of convex bodies admitting invariant hypersurfaces for the symplectic outer billiards map, each consisting of 4-periodic centrally symmetric orbits.

Main result

Theorem. The space of symplectically self-polar bodies of a given dimension is contractible.

The proof combines two structural facts about the class with an explicit isotropic rotation. The ingredients are:

• Symplectic $\ell_2$-sum: for bodies $X, Y$ with associated Minkowski functionals, $X \oplus_2 Y$ is defined by the condition $\|x\|_X^2 + \|y\|_Y^2 \leq 1$ on $\mathbb{R}^{2n} \times \mathbb{R}^{2n}$, viewed as a direct sum of symplectic spaces. The key identity $X = X^\omega$0 implies that the $X = X^\omega$1-sum of two self-polar bodies is again self-polar.
• Linear symplectic reduction: for an isotropic subspace $X = X^\omega$2, the reduction $X = X^\omega$3 commutes with symplectic polarity, so reductions of self-polar bodies remain self-polar. The paper includes a self-contained proof via Hahn–Banach extension of a linear functional of the form $X = X^\omega$4, constructing a preimage vector $X = X^\omega$5 whose projection recovers $X = X^\omega$6.

Given two self-polar bodies $X = X^\omega$7 and $X = X^\omega$8, the proof forms $X = X^\omega$9 in $\omega$0 and considers the one-parameter family of isotropic subspaces

$\omega$1

where $\omega$2 and $\omega$3 are standard symplectic bases of the two factors. Each quotient $\omega$4 carries a canonical symplectic basis, giving identifications $\omega$5 with $\omega$6. The reduced bodies

$\omega$7

are all symplectically self-polar by the reduction lemma, depend continuously on $\omega$8, and satisfy the endpoint identities $\omega$9 and $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$0. Fixing any basepoint body $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$1 therefore yields a contraction of the entire space to that point.

An immediate consequence worth noting: since the space is path-connected, there are no topological obstructions to deformation arguments within the class — for instance, variational or continuity methods over the space of self-polar bodies need not track components.

Planar examples

The paper illustrates the homotopy explicitly in dimension two for two pairs of extremal bodies. Planar self-polar hexagons are distinguished: up to linear symplectomorphism, the hexagon is the unique planar self-polar body of minimal volume, while the Euclidean disk is the unique planar self-polar body of maximal volume, following from the equality case in the Blaschke–Santaló inequality.

For the pair of hexagons with norms $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$2 and $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$3, the interpolating family $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$4 is computed exactly: it consists of the self-polar hexagons

$$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$5

with membership verified by evaluating the defining quadratic constraint at three vertices and using central symmetry together with inclusion-reversal of symplectic polarity.

For the ball–hexagon pair, the interpolants $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$6 are no longer polytopes; their boundaries are piecewise given by arcs of ellipses and conics determined by explicit functions $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$7. The verification that $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$8 proceeds by checking that each boundary arc admits a witness parameter $$X^\omega = \{y \in \mathbb{R}^{2n} \mid \forall x \in X,\ \omega(x,y) \leq 1\}.$$9 realizing the constraint $\vol X \geq 2^n/n!$0, then concluding equality via polarity reversal as before. These computations confirm that the abstract homotopy produces geometrically transparent families connecting even the volume-minimizing and volume-maximizing planar members of the class.

Limitations and open questions

The result concerns only the coarse topology of the class; it does not address finer geometric structure such as metric properties of the Hausdorff space, stratification by combinatorial type, or behavior under additional regularity constraints. The author also notes that while a sequence of self-polar polytopes $\vol X \geq 2^n/n!$1 minimizing the Ekeland–Hofer–Zehnder capacity has been constructed, and it "seems" that each $\vol X \geq 2^n/n!$2 minimizes volume among self-polar bodies in $\vol X \geq 2^n/n!$3, this volume-minimality remains unproven in general dimensions — precisely the content of the self-polar reformulation of Mahler's conjecture. Whether the contractibility established here can be leveraged in a minimax or deformation argument toward that volume inequality is left open.

Conclusion

The paper proves that the space of symplectically self-polar convex bodies of fixed dimension is contractible, via an elementary but effective construction combining the symplectic $\vol X \geq 2^n/n!$4-sum with a rotating family of isotropic reductions. Together with the explicit planar interpolations between extremal bodies, the result supplies the foundational topological picture for this class and clarifies that its richness lies entirely in geometry rather than topology.